Find the square of 527 by using the diagonal method.

Answer

To find the square of 527 using the diagonal method, we will follow these steps:

### Step-by-Step Solution:

1. **Set Up the Grid**:
- Since we are squaring a three-digit number (527), we will create a 3x3 grid, which consists of 9 parts.

2. **Write the Number**:
- Write the number 527 along the top and left side of the grid. It will look like this:
```
| 5 | 2 | 7 |
-----------------
5 | | | |
-----------------
2 | | | |
-----------------
7 | | | |
```

3. **Multiply Each Pair**:
- Fill in the grid by multiplying the corresponding digits:
- For the cell (5,5): \(5 \times 5 = 25\)
- For the cell (5,2): \(5 \times 2 = 10\)
- For the cell (5,7): \(5 \times 7 = 35\)
- For the cell (2,5): \(2 \times 5 = 10\)
- For the cell (2,2): \(2 \times 2 = 4\)
- For the cell (2,7): \(2 \times 7 = 14\)
- For the cell (7,5): \(7 \times 5 = 35\)
- For the cell (7,2): \(7 \times 2 = 14\)
- For the cell (7,7): \(7 \times 7 = 49\)

- The grid now looks like this:
```
| 5 | 2 | 7 |
-----------------
5 | 25 | 10 | 35 |
-----------------
2 | 10 | 4 | 14 |
-----------------
7 | 35 | 14 | 49 |
```

4. **Add the Diagonals**:
- Now, we will add the numbers along the diagonals:
- First diagonal (bottom left to top right): \(49\)
- Second diagonal: \(35 + 14 + 10 = 59\)
- Third diagonal: \(25 + 10 + 4 = 39\)
- Fourth diagonal: \(35 + 10 = 45\)
- Fifth diagonal: \(25\)

5. **Combine the Results**:
- Now we will combine these results, starting from the bottom diagonal to the top:
- From the last diagonal, write down \(9\) (from \(49\))
- Next, \(5 + 9 = 14\) (write down \(4\) and carry \(1\))
- Next, \(3 + 5 + 1 = 9\) (write down \(9\))
- Next, \(2 + 1 = 3\) (write down \(3\))
- Finally, write down \(2\) from the first diagonal.

6. **Final Result**:
- Putting it all together, we get \(277729\).

### Conclusion:
The square of 527 using the diagonal method is **277729**.